1. Field of the Invention
This invention relates generally to computer-based modeling of gravity hydraulic network systems and, more particularly, to techniques for a representation of flow data within such systems.
2. Background Information
Computer-based modeling solutions for simulating the complete hydrodynamic response of pipe hydraulic network systems, such as those encountered in storm water and sanitary sewer collection systems are useful in the design, rehabilitation and simulation of such systems. The software, which performs the functions that describe the water distribution network, is often referred to as an hydraulic network solver or hydrodynamic solver (often commonly referred to generally as a “solver”). The solver is a computer program that simulates and predicts water pipe flows, hydraulic pressure conditions, run-off water, storm conditions and flooding conditions. In a network of pipes, the conditions can be described for the links (the pipes) and the nodes (the junctions) in such a system. There are a number of available software products that embody hydraulic network solvers and that provide general-purpose modeling.
The solvers, inter alia, compute and provide hydraulic solutions for complex differential equations. More specifically, computer-based modeling solutions for simulating the complete hydrodynamic response of hydraulic network systems, such as those encountered in storm water and sanitary sewer collection systems, are typically implemented employing numerical techniques for solving St. Venant equations. Frequently, solutions are obtained by iteratively solving a simultaneous system of equations governing the implicit solution of a finite-difference scheme that describes variation of flow over a mathematical grid of time and one dimension space. These equations that govern the system-wide hydraulic response are non-linear and are most often solved using Newton-Raphson numerical matrix-based solvers. The matrix is typically comprised of a set of equations that each describe a point on the mathematical grid. The results produced may include a family of hydraulic delivery curves that describe various aspects of a storm water and sanitary system.
In practice, it is quite difficult to implement a general modeling solution that exhibits accuracy and robustness over the entire range of flow conditions (and over the transitions that occur between these flow conditions) that are of interest to the engineering modeler. For example, in man-made hydraulic networks, flows diverge at junctions or man-made diversions and then outfall from the system entirely or split off and then return to converge again with the system at some down stream location. These transitions are difficult to model and it is difficult to thus provide hydraulic response curves for such aspects of the system. Other examples of transitions include those occurring when flows transition between dry-bed to free surface, or from gravity flow to pressurized flow, or from steeply sloped channels to mildly sloped, or flat, or adversely sloped, channels. In addition, a transition can occur from pipe to manhole to ditch to culvert. Each of these flow conditions, flow transitions and hydraulic situations needs to be described by a different set of governing equations. Successfully employing conventional numerical techniques with the variety of mathematical formulations that describe the various hydraulic structures is a challenging task.
More specifically, conventional techniques have run in to difficulty when attempting to describe the numerous hydraulic structures providing these transitions. Up to now, because the mathematical description of these systems is so complex, the trend has been to limit the range and type of network flow components used in the model to those elements whose hydraulic response can be described by simple equations, such as those occurring in pipe, manholes, weirs, orifices and prismatic channels. Thus, the conventional network solvers typically do not incorporate into the model such components as culverts, siphons, and detention pond control structures because they cannot be described mathematically by a single unified equation. However, to simply ignore those components does not give rise to an accurate model or a useful family of curves. Thus, approximations are often used to attempt to account for these structures that do not lend themselves to conventional numerical techniques, but these approximations are not always accurate.
Moreover, many models simulate network behavior at a particular time period, but the characteristics of the actual system continuously change with time. In addition, the equations used typically run in to difficulty during transition times. Transitions can result in separate, more complex transitional equations that are not readily solved by the solver engine. And if solved, these equations can provide unreliable data at transition points. This typically occurs in gravity networks. At transitional points, the equation results can also become unstable and are not robust.
The complexities are also increased when a number of different structures are involved in the solution. More particularly, when a structure or component is involved in a network analysis, that structure is typically described by a mathematical protocol that consists of selecting a particular set of working equations that are invoked depending upon the changing conditions that occur over the range of operation for that structure. For example, a typical pond riser flow response subject to rising flow will be governed first by weir limited flow, then entrance limited culvert flow and then partially submerged culvert flow and then, ultimately, submerged flow. Thus, four different sets of working equations must be employed to develop that flow response in that instance.
The hydraulic structures mentioned are typically dominated by local head losses and are transitional in nature. In storm-water management systems, these structures are frequently used to limit or control flow so that down stream areas are not severely impacted by unmanaged discharges from disturbed or developed areas, which could result in flooding conditions. For this reason, it is important to accurately model the hydraulics of these structures to determine their effectiveness, and thus their safety and effectiveness for protecting the community and the environment. However, the complex mathematical representation of such structures generally precludes their integration into conventional hydrodynamic network solvers, and instead engineers must utilize alternative modeling approaches to design or analyze systems that contain these structures, thus resulting in increased time and cost of the analysis.
Such alternative modeling approaches include employing conservative approximations and simplifying assumptions in order to model systems that include such structures. This results in a risk of oversimplification and inaccurate solutions describing the model. Alternatively, an intensely iterative explicit solution technique could be employed, but these can be slow to converge, resulting in lengthy run times that can hinder the efficiency of the design process and result in increased costs.
Due to these difficulties, the mathematical descriptions of these systems can involve three-dimensional surfaces that are not smooth, and in other words cannot be differentiated. The data that is produced for flow (O) as a function of tailwater (TW) and headwater (HW) includes data that is comprised of three matrices of values: TWi, HWj, Qij for 0≦i≦m and 0≦j≦n where Qij is the flow which corresponds to tailwater (TWi) and headwater (HWj). At transitional points, the three-dimensional surface representing the three matrices of data can be non-smooth and undifferentiable. Thus, there remains a need to produce a technique for providing a triangulated surface interpolation to the data points, which maintains the monotonicity of the data and allows for a quick evaluation of a flow value for any point within the system. There remains a further need for a smooth surface representation of flow data which provides a more detailed and accurate representation of the flow data.
There remains a further need for a straight-forward method and system for generating these surfaces that allows for the description of flow characteristics, transitional-flows or transitional points and other flow limiting and flow controlling structures in a hydraulic network.
It is thus an object of the present invention to provide a software program that generates highly accurate and robust solutions to model any passive hydraulic structure within a hydraulic network by providing the triangulated surface and smooth surface interpolation techniques.
It is a further object of the invention to provide a system and method for generating solutions that allow hydraulic modelers to accurately incorporate transitional hydraulic elements into a conventional, hydrodynamic network solver without lengthy computations or a variety of different sets of equations.